I have a question about my method of proof for proving a simple fact about projective modules. I have a feeling my idea is wrong and I was hoping some one could point out where the mistake is. Let $R$ be a commutative ring with identity. Let $P$ be a projective $R$-module. Prove that there […]

Did anyone find a proof of this theorem? I can’t find it on the Internet. The theorem is : Let $X$ be a set and let $R$ be a set of reduced words on $X$. Assume that a group $G$ has the presentation $\langle X \mid R \rangle$. If $H$ is any group generated by […]

I am trying to understand the partitions of $S_5$ created by it’s conjugacy classes but two sources have two different partitions. Source 1: Source 2: So, for example, in the first table, the partition for cycle structure ()()()()() i.e. $5$ $1$-cycle is $5+0+0+0+0$ but in the second table it is $1+1+1+1+1$. Could anyone please help […]

Looking at the definition of a monoid it says that: A monoid is a set that is closed under an associative binary operation and has an identity element $I \in S$ such that for all $a \in S$, $I a = a I =a$ But what does $I a$ mean here? I mean it’s just […]

Take $R = \mathbb{Z}$ and fix a set of nonzero integers $X = \{n_{1},n_{2},\dots, n_{k} \}$. Now, let $$D = \{n_{1}^{t_{1}}n_{2}^{t_{2}}\cdots n_{k}^{t_{k}} \mid t_{i}\geq 0 \, \text{for all}\, i,\ t_{i}\neq 0\, \text{for at least one}\, i\}$$ i.e., $D$ is a semisubgroup generated by $X$. I need to show that $D^{-1}\mathbb{Z}$ is isomorphic to another ring […]

As the question suggests, is the cuspidal curve $\mathcal{M}$ a coarse moduli space for lines in $\mathbb{C}^2$? I’m inclined to believe the answer is no, but all attempts at proving it so far have seemed not fruitful…

The definition of a normal extension in the book “Abstract algebra” is : If $K$ is an algebric extension of $F$ which is the splitting field over $F$ for a collection of polynomials $f(x)\in F[x]$ then $K$ is called a normal extension I think that there is something here I don’t understand: If $K$ is […]

Given a field $F$ and a polynomial $P \in F[x]$ such that $P$ is irreducible over $F$. Let $L_P$ be the splitting field of $P$ and $F$. Does $\operatorname{dim}_F({L_F}) = \deg(P)$ hold? If looking for a the minimal polynomial of $\alpha$ in a field $F$ is it sufficient to find a polynomial $P$ which is […]

First, suppose $i_1$ and $i_2$ are additive identity in ring R. From the definition of “additive identity” $a+i_1=a$ such that there is $a$ $\in$ $R$, including for $a=i_2$, so $i_2+i_1=i_2$. But $i_2$ is also an additive identity, so there $a \in R$ making $a+i_2=a$. Then $a=i_1$. Then $i_1+i_2=i_1$. Since this is a ring, addition is […]

The problem consists of two parts: 1)Prove that for some integer $m,$ $3^{m} = 1$ in the integers modulo 7, $\mathbb{Z}_{7}$, which i have done. The second part asks to use this fact to deduce that $7$ divides $1 + 3^{2001}.$ The way i look at it, we look at $1 + 3^{2001}$ in $\mathbb{Z}_{7}$ […]

Intereting Posts

Advice about taking mathematical analysis class
Contractions mappings bijective maps boundarys on boundarys?
Countable family of Hilbert spaces is complete
Could I see that the tensor product is right-exact using its universal property and the Yoneda lemma?
Did I derive a new form of the gamma function?
Reflected rays /lines bouncing in a circle?
Field axioms: Why do we have $ 1 \neq 0$?
Is calculating the summation of derivatives “mathematically sound”?
How to find the maximum diagonal length inside a dodecahedron?
when is $\frac{1}{n}\binom{n}{r}$ an integer
Applications of the “soft maximum”
What is “ultrafinitism” and why do people believe it?
An incorrect method to sum the first $n$ squares which nevertheless works
A logic that can distinguish between two structures
If $X^\ast $ is separable $\Longrightarrow$ $S_{X^\ast}$ is also separable